#6989777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,7,8,5)(9,10,14,13)(11,15,12,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,33)(28,40)(29,38)(31,45)(32,46)(34,47)(35,42)(36,48)(39,44)(41,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,28,32,33)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(3,8)(5,7)(9,12,14,11)(10,15,13,16)(17,20,19,28,23,32,22,33)(18,25,37,46,27,40,24,30)(21,34,29,44,35,31,36,43)(26,41,48,45,42,39,38,47);(2,6)(5,7)(9,10,14,13)(11,15,12,16)(17,21,23,35)(18,26,27,42)(19,29,22,36)(20,31,32,34)(24,38,37,48)(25,39,40,41)(28,43,33,44)(30,45,46,47);(9,13,14,10)(11,15,12,16)(20,32)(21,35)(25,40)(26,42)(28,33)(29,36)(30,46)(38,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896973);